Step |
Hyp |
Ref |
Expression |
1 |
|
climcncf.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
climcncf.2 |
|- ( ph -> M e. ZZ ) |
3 |
|
climcncf.4 |
|- ( ph -> F e. ( A -cn-> B ) ) |
4 |
|
climcncf.5 |
|- ( ph -> G : Z --> A ) |
5 |
|
climcncf.6 |
|- ( ph -> G ~~> D ) |
6 |
|
climcncf.7 |
|- ( ph -> D e. A ) |
7 |
|
cncff |
|- ( F e. ( A -cn-> B ) -> F : A --> B ) |
8 |
3 7
|
syl |
|- ( ph -> F : A --> B ) |
9 |
8
|
ffvelrnda |
|- ( ( ph /\ z e. A ) -> ( F ` z ) e. B ) |
10 |
|
cncfrss2 |
|- ( F e. ( A -cn-> B ) -> B C_ CC ) |
11 |
3 10
|
syl |
|- ( ph -> B C_ CC ) |
12 |
11
|
sselda |
|- ( ( ph /\ ( F ` z ) e. B ) -> ( F ` z ) e. CC ) |
13 |
9 12
|
syldan |
|- ( ( ph /\ z e. A ) -> ( F ` z ) e. CC ) |
14 |
1
|
fvexi |
|- Z e. _V |
15 |
|
fex |
|- ( ( G : Z --> A /\ Z e. _V ) -> G e. _V ) |
16 |
4 14 15
|
sylancl |
|- ( ph -> G e. _V ) |
17 |
|
coexg |
|- ( ( F e. ( A -cn-> B ) /\ G e. _V ) -> ( F o. G ) e. _V ) |
18 |
3 16 17
|
syl2anc |
|- ( ph -> ( F o. G ) e. _V ) |
19 |
|
cncfi |
|- ( ( F e. ( A -cn-> B ) /\ D e. A /\ x e. RR+ ) -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) ) |
20 |
19
|
3expia |
|- ( ( F e. ( A -cn-> B ) /\ D e. A ) -> ( x e. RR+ -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) ) ) |
21 |
3 6 20
|
syl2anc |
|- ( ph -> ( x e. RR+ -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) ) ) |
22 |
21
|
imp |
|- ( ( ph /\ x e. RR+ ) -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) ) |
23 |
4
|
ffvelrnda |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. A ) |
24 |
|
fvco3 |
|- ( ( G : Z --> A /\ k e. Z ) -> ( ( F o. G ) ` k ) = ( F ` ( G ` k ) ) ) |
25 |
4 24
|
sylan |
|- ( ( ph /\ k e. Z ) -> ( ( F o. G ) ` k ) = ( F ` ( G ` k ) ) ) |
26 |
1 2 6 13 5 18 22 23 25
|
climcn1 |
|- ( ph -> ( F o. G ) ~~> ( F ` D ) ) |